Isosceles right triangle $ABC$ encloses a semicircle. The circle has its center $O$ on hypotenuse $\overline{AB}$ and is tangent to sides $\overline{AC}$ and $\overline{BC}$. If the length of the hypotenuse $\overline{AB}$ is $4\sqrt{2}$, what is the area of the semicircle?